Manley-Rowe relations

The Manley-Rowe relations state that in a lossless second-order (three-wave) interaction among waves at $\omega_1$, $\omega_2$ and $\omega_3 = \omega_1 + \omega_2$, the photon fluxes change in lockstep: every photon created at $\omega_3$ removes one photon from each of the lower-frequency waves, and every photon split from $\omega_3$ adds one to each. Since a photon carries energy $\hbar\omega$, the power exchanged with each wave is proportional to its frequency. In difference-frequency generation of 3.39 µm from 1064 nm and 1550 nm, at most $1064/3390 = 31\%$ of the pump power can reach the idler; in sum-frequency generation of 630.9 nm from the same pair, each watt of 1550 nm light fully converted emerges as 2.46 W.

Statement

In terms of the intensities $I_j$ (W/m²) along the propagation direction $z$,

$$\frac{d}{dz}\!\left(\frac{I_1}{\omega_1}\right) = \frac{d}{dz}\!\left(\frac{I_2}{\omega_2}\right) = -\frac{d}{dz}\!\left(\frac{I_3}{\omega_3}\right).$$

Each quotient $I_j/\omega_j$ is, up to the factor $\hbar$, a photon flux per unit area. Two independent conserved combinations follow, for example $I_1/\omega_1 + I_3/\omega_3$ and $I_2/\omega_2 + I_3/\omega_3$, and adding the three intensities recovers conservation of total power, $I_1 + I_2 + I_3 = \text{constant}$.

Derivation outline

In the slowly varying envelope approximation the field amplitudes obey coupled equations of the form

$$\frac{dA_1}{dz} = \frac{2i\,d_\text{eff}\,\omega_1}{n_1 c} A_3 A_2^{*} e^{i\Delta k z},$$

with matching equations for $A_2$ and $A_3$, where $d_\text{eff}$ is the effective nonlinear coefficient and $\Delta k$ the phase mismatch. Each intensity is $I_j = 2n_j\varepsilon_0 c\,|A_j|^2$. Forming $dI_j/dz$ from these equations gives, for every wave, the frequency $\omega_j$ multiplied by one common quantity (with opposite sign for $\omega_3$), and dividing by $\omega_j$ yields the relations above. The derivation needs only that the medium is lossless, so that $\chi^{(2)}$ has full permutation symmetry; it holds with or without phase matching, which affects how fast power flows but not the ratio in which it is shared.

The quantum-defect limit

Because a pump photon at $\omega_3$ splits into one signal and one idler photon, an optical parametric oscillator or amplifier with complete pump depletion delivers the fraction $\omega_s/\omega_p$ of the pump power to the signal and $\omega_i/\omega_p$ to the idler. For a 1064 nm pump, a 1550 nm signal and a 3.39 µm idler these are 69% and 31%. The remainder leaves the crystal in the signal wave. The idler fraction is the parametric analog of the quantum defect in laser pumping, and it is the reason that mid-infrared parametric sources pumped in the near-infrared have low power conversion even at perfect photon conversion.

In second-harmonic generation two photons at $\omega$ produce one at $2\omega$, so the power conversion efficiency can in principle reach 100%. In sum-frequency generation the weaker input limits the photon flux: 1 W at 1550 nm carries 7.80 × 10¹⁸ photons per second, and converting all of them to 630.9 nm requires the 1064 nm beam to supply at least the same photon flux, which is 1.46 W at 1064 nm.

Where it matters

The relations set the ceiling for wavelength conversion efficiency, determine how much pump must be depleted in a parametric amplifier to reach a given signal gain, and explain gain in parametric up-conversion: a weak infrared signal converted to the visible by sum-frequency mixing gains power by the ratio $\omega_3/\omega_1$ while keeping its photon number. They were derived originally for nonlinear reactances in microwave circuits and apply to any lossless parametric process.

Pitfalls

Absorption at any of the three wavelengths, competing processes such as parasitic second-harmonic generation, and two-photon absorption break the simple photon accounting. Quoted efficiencies should state whether they refer to power or photon number, since the two differ by the frequency ratio.

Common questions

Do the Manley-Rowe relations require phase matching?

No. They follow from the form of the coupled equations for any phase mismatch. Phase mismatch makes the energy flow back and forth over a coherence length, but at every point the photon fluxes still change in the fixed ratio.

Why can difference-frequency generation amplify one of the input beams?

Each difference-frequency photon at $\omega_3 - \omega_2$ is created together with a photon at $\omega_2$, while one photon at $\omega_3$ is consumed. The lower-frequency input therefore gains the same photon flux as the idler, which is optical parametric amplification.

References: J. M. Manley and H. E. Rowe, "Some general properties of nonlinear elements. Part I. General energy relations," Proc. IRE 44, 904 (1956); R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).